You should not be in a situation where both checks apply
Rolling lower than your passive score would indeed be a conceptual problem, doing worse than what your minimum should be, if you were ever in a situation where you could use both kinds of check. But, by design, the two kinds of checks are used at different times, in different situations, so there is not a conflict. When one kind of check is being used, the other should not be.
In the case that Passive checks are used, they are for two completely different reasons, as explained in the PHB (emphases mine):
A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster...The rules on hiding in the “Dexterity” section below rely on passive checks, as do the exploration rules.
Taking the second use case first, since it is the one presented by the OP, Passive checks are used to contest Stealth specifically when the PC's are looking for hidden monsters. This is the general case when the DM does not want the players to know the results of rolls, often because the characters would not know the results of the attempt.
This is in contrast to a situation in which the characters will immediately know whether their attempt succeeded or failed. If I make an Athletics check to "jump an unusually long distance", there is no meta-information in the roll, because my character already knows whether they succeeded or failed. If I am looking for Hidden foes, however, and do not find them, I should not know whether I looked poorly and failed, or looked well but no foes were there.
Consider the following scenario - the party scout checks a colonnaded hall, where opportunities for Hidden foes abound. If the DM were to allow a Perception roll, the player would know what their character got, and metagaming could ensue. 'I rolled a 19, so I'm sure there's nothing here, let's move on.' vs. 'I rolled a ten - let's proceed cautiously, since there could be Hidden foes here.' Or, as they say in 3.5e, 'I think I just failed a Spot check.'
The point is that the player should not know 'how well they perceived' if they are then going to act on that meta-information. It is assumed that their character is always looking to the best of their ability, but doesn't know whether they failed or by how much they succeeded. Rather, they know they are a 'good scout' (based on their Passive score), and that they will see anything obvious or typically Hidden - but they have to assume that a very Stealthy foe will escape their notice. So when the DM says 'the hall appears empty', the player does not know whether that means there is actually nothing there, or that the things that are there are just very good at Hiding - which is precisely the information their character has.
In this case, it is not possible to roll 'under the floor' of the passive Perception, since if the DM is calling for passive Perception it is because they specifically do not want rolls to be made.
The other use case for passive Perception is for the average of 'a task done repeatedly'. Here, the DM would determine what 'repeatedly' means, and use a passive score in some cases, or call for a roll in others, but not both in the same situation.
Suppose the characters come upon a closed door. At this point they are being cautious in their explorations and the scout declares that they are going to listen at the door for several minutes. The DM decides that this is a task done repeatedly, and uses the scout's passive Perception to tell them what they hear - the muted conversation of the monsters within. There is no chance of 'rolling under the floor', because as soon as the DM determines that a passive score will be used, no roll is going to be made.
Now take the same characters, the same door, and the same monsters within. This time, however, the characters are being pursued and are just a few rounds ahead of their pursuers. Now the scout says 'I listen at the door, but if I don't immediately hear something, I open it and continue moving'. In this case the DM calls for a roll - the scout is spending their combat-round action on a Perception check. They might roll high and hear the monsters, or they might roll low and not. It would be unlucky for them to be listening during a six-second lull in the monster's conversation, but it could happen. If it did, they would not be rolling under their floor, because their floor presumes repeated attempts that average over time. In this case, they just get one shot, and sometimes that means they underperform compared to their typical ability.
Some characters "have refined [their] chosen skills until they approach perfection." For them, they are so reliably good, it is simply impossible for them to be very unlucky, even at a one-shot chance of something. In this case, their passive score is equivalent to the floor to their roll, but it is not the ceiling, because the DM has called for a roll, and there is a fair chance they will roll higher than the average. As Groody points out in his answer, this is the case for Rogues with Perception as their Reliable Talent. As the exception that proves the rule, assigning this class feature to this skill is valuable only because other characters can roll under what would be their passive 'floor' (were Passive scores allowed) in the same situation.
There is no conflict when a low Perception roll is under the passive Perception, because rolled and passive Perception should never occur in the same circumstances.